Notes 10.7 – Polar CoordinatesRectangular Grid

(Cartesian Coordinates)Polar Grid

(Polar Coordinates)

Origin Pole

Positive X-axisPolar axis

There is no equivalent to the y-axis in polar coordinates.

Notes 10.7 – Polar CoordinatesRectangular Grid

(Cartesian Coordinates)Polar Grid

(Polar Coordinates)

Point (x, y) Point (r, )x = Horizontal Distance

The point doesn’t move.The point doesn’t move. We are just changing how the point is labeled.

y = Vertical Distance

r = Radial Distance(outward from pole)

= Angle from polar axis(measured counterclockwise)

0

The angles, in degrees………90

180

270

, 360

45

30

6075

15

Polar graphs can come in different

increments.For this one, there

are 6 steps to get to the 90 location so each step would be

90/6 or 15

105120

165

150

135

225

210

240255

195

285300

345

330315

32

412

The angles, in radians………

0

2

, 2

Polar graphs can come in different

increments.For this one, there

are 6 steps to get to the /2 location so each step would be

(/2)/6 or /12

12

512

312

212

Since the next one would be 6/12 which is /2, we should feel comfortable with our

counting.

If you kept going, and reduced your

fractions, you would have……

Ex. 1) Plot the point 4,4

Find /4. Here it is!!

Go out 4 from the pole and plot the point.

Alternate labels…… 4,4

This point could stay in the same place but be labeled differently.

If I go completely around the circle, the point would be

94,4

If I go around the circle the other way, the point would be

74,4

Alternate labels…… 4,4

You can also look at it this way…...

If I go out on the 5/4 line but go in a negative direction, the point would be

54,4

The point never moved, it was just labeled differently.

In rectangular coordinates, there is not this issue. There is only one way to label each point. In polar, we can label a point an infinite number of ways. Fortunately, MOST of the time, we only look at values of between 0 and 2.

Ex 2) Plot the following points: 53,0 4, 5, 6,

3 4 4A B C D

DC B

A

Converting from Polar to Rectangular Coordinates

Remember from your basic trig:• x = r cos • y = r sin

Ex 3) Convert from Polar to Rectangular Coordinates

) 2,6

b

) 4,3

a

cos14cos 4 2

3 2

x r

x

sin

34sin 4 2 33 2

y r

y

So the rectangular coordinates of the point would be: 2,2 3

cos

32cos 2 36 2

x r

x

sin

12sin 2 16 2

y r

y

So the rectangular coordinates of the point would be: 3,1

The point doesn’t move……Take the point from part a of the previous example and plot it on the polar grid.

Note where the pole is.

If we bring in the rectangular grid so that the pole & origin line up (as they should)…..

Then you see the point on the rectangular grid in the same location.

…at least as well as your teacher can line up the grids!!

Converting from Rectangular to Polar Coordinates

Remember from your basic trig:

tan yx

In conjunction with the Pythagorean Theorem:

2 2 2r x y

Use these to convert (x, y) to (r, ).

Ex 4) Convert the following from Rectangular to Polar Coordinates

a) (0, 3)2tan 12

Plot the point on

a polar grid as if it were rectangular. You will see that the polar coordinates are:

22 22 2r

b) (2, -2) ) 3,1c

3,2

1tan 1 4

2 2r

So the polar coordinateswould be:

72 2, 2 2,4 432 2,4

or

or

1 3tan33

1 3tan3

6

But that angle would be in the 4th quadrant and our point is in the 2nd quadrant. Thus we must add .

22 23 1r 2r

56

So the polar coordinateswould be: 52,

6

Steps for Converting from Rectangular to Polar Coordinates

1. Always plot the (x, y) point.2. Find r using:3. Find using: 1tan y

x

If x, y is in Quadrant I, is OK.If x, y is in Quadrant II or III, add to get in the correct location.If x, y is in Quadrant IV, add 2 to get in the correct location and between 0 and 2.

2 2 2r x y

Transforming EquationsPolar to Rectangular

Ex 5) Transform r = 6 cos from polar to rectangular form.

r = 6 cos

Since our conversions don’t involve r or just cos but rather r2 and r cos , multiply both sides by r to get the correct format.

r2 = 6r cos Now use the same conversions as earlier with points.

r2 = 6r cos x2 + y2 = 6x

Ex 5) Continued……

Since this is the equation of a circle, we can convert to standard form.

Center (3, 0) with a radius of 3.

(x - 3)2 + y2 = 9

x2 + y2 = 6x

x2 – 6x + y2 = 0x2 – 6x + 9 + y2 = 0 + 9

Transforming EquationsRectangular to Polar

Ex 6) Convert 4xy = 1 to a polar equation.Solve for r or r2, if possible.

Use the same conversions as earlier with points.

4xy = 14(r cos )(r sin ) = 1

4r2 cos sin = 1

2r2 sin 2 = 1

r2 = ½ csc 2